158 ANALYTICAL MECHANICS ma2 . T7ra2Z3 4 12 3. Find the moment of inertia of a homogeneous sphere about a diameter and about a tangent line. Let m, a, and r be the mass, the radius, and the density of the sphere, respectively. Then, taking the axes and the element of mass as shown in Fig. 89, we have dly = dly" + 22 dm, Integrating the last equation s*m f* Iy^ Uo y2dm+ Jo = V Cy4 dz + TTT 4 •'—a dz [dm = Tiry*dz] ~ 15 and | ma2. 124. Theorem III. — The moment of inertia of a homogeneous right cylinder about a transverse axis equals the moment of inertia of two lamince which fulfill the following conditions, (a) Each lamina has a mass equal to that of the cylinder. (6) One lamina occupies the entire area of the transverse section of the cylinder through the given axis, while the other lamina occupies the entire area of the longitudinal section of the cylinder through the axis.